102 XIII INTEGRATION and Supp(/£)c Uar For this purpose we observe that if K is the support of /, then there exist finitely many indices af e I (1 ^ i :g ri) such that the Uaf cover K. Hence ((3.18.2) and (12.6.4)) there exist n continuous mappings ht'.X-* [0, 1] such that Supp(Af) is compact and is contained in Uai for n 1 <£ i ^ 72, and such that £ Af(jc) = 1 for all x e K. Then the functions/; =/A, »=i satisfy the required conditions. This already proves the uniqueness of f* : for by definition we must have To prove the existence of a linear form ft on ,3rc(X) whose restriction to jTc(Ua) is /ia for each a e I, it is enough to establish the following assertion: given two finite sequences (ffi)i^i^m and (hj)i£jgn of functions belonging to JTC(X), such that Supp(#f) c U~ for l^i^m and Supp^) c U^^ for 1 ^j ^n, and such that for all x e Supp(/), then we have 1-1 /=! Now, we have and therefore m m / n \ E^«X/^-)= Z T,H*t(f9ihj) • i=J i=l\J=l / Similarly, But since Supp(/^f/i7.) is contained in UXi n Uftj, it follows from the hy- potheses that ^ai(fyihj) = /*/»//<?!/*/), and our assertion follows. It remains to be shown that the linear form \i so defined is a measure. Let K be a compact subset of X, and define the Uai and the ht as at the be- ginning of the proof. If Hf = Supp(Af), then by hypothesis there exists an write /in place of/| U when Supp(/) c U, and g in place of #u). The